Log-Concave Duality in Estimation and Control

In this paper we generalize the estimation-control duality that exists in the linear-quadratic-Gaussian setting. We extend this duality to maximum a posteriori estimation of the system's state, where the measurement and dynamical system noise are independent log-concave random variables. More generally, we show that a problem which induces a convex penalty on noise terms will have a dual control problem. We provide conditions for strong duality to hold, and then prove relaxed conditions for the piecewise linear-quadratic case. The results have applications in estimation problems with nonsmooth densities, such as log-concave maximum likelihood densities. We conclude with an example reconstructing optimal estimates from solutions to the dual control problem, which has implications for sharing solution methods between the two types of problems

Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finite-dimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems

Global Convergence of Policy Gradient Primal-dual Methods for Risk-constrained LQRs

While the techniques in optimal control theory are often model-based, the policy optimization (PO) approach can directly optimize the performance metric of interest without explicit dynamical models, and is an essential approach for reinforcement learning problems. However, it usually leads to a non-convex optimization problem in most cases, where there is little theoretical understanding on its performance. In this paper, we focus on the risk-constrained Linear Quadratic Regulator (LQR) problem with noisy input via the PO approach, which results in a challenging non-convex problem. To this end, we first build on our earlier result that the optimal policy has an affine structure to show that the associated Lagrangian function is locally gradient dominated with respect to the policy, based on which we establish strong duality. Then, we design policy gradient primal-dual methods with global convergence guarantees to find an optimal policy-multiplier pair in both model-based and sample-based settings. Finally, we use samples of system trajectories in simulations to validate our policy gradient primal-dual methods